QuantumPhysics.dvi

not primordial. This lack of exact symmetry of orbits is what ultimatelycaused the downfall

of the Ptolemy model of the planets.

By definition,a symmetry of an equationis atransformation on the dynamical variables

that maps any solution to the equation to a solution of the same equation. Symmetry

transformations may be

• discrete

(such as parity, time reversal, translation and rotation in a crystal);

• continuous, i.e. parametrized by continuous parameters

(such as translation and rotation invariance in the continuum).

It is well-known that time and space translation invariance imply conservation of energy

and momentum respectively and that rotation invariance implies conservation of angular

momentum. This connection between a continuous symmetry and anassociated conserved

quantity orconserved chargeis a general one thanks to a theorem by Emmy Noether.

We consider a mechanical system described by a LagrangianL(q,q ̇;t) for positionsqi(t)

withi = 1,···,N. Now consider a continuous symmetry acting onqi(t). A continuous

symmetry (such as a translation or a rotation) will depend upon a continuous parameterα,

which we shall assume to be real. Thus, the new positions ̃qiare given as a function of this

parameter, in such a way that

q ̃i(t,α) = differentiable function ofα

q ̃i(t,0) = qi(t) (9.1)

For any givenα, the transformationqi(t)→q ̃i(t,α)is a symmetry provided every solution

qi(t)of the Euler-Lagrange equations is mapped into a solutionq ̃i(t,α)of the same Euler-

Lagrange equations. In particular, this means that ifqi(t) is a stationary trajectory of the

actionS[q], then so is ̃qi(t,α). (Notice that the values of the end pointsqi(t 1 , 2 ) will in general

be transformed into different valuesqi(t 1 , 2 ,α).) Since the symmetry is continuous, we may

restrict to an infinitesimal version of the transformation, definedby

δqi(t) =δqi(q,q ̇;t) = limα→ 0

q ̃i(t,α)− ̃qi(t,0)

α

=

∂q ̃i(t,α)

∂α

∣∣

∣∣

α=0

(9.2)

The infinitesimal transformationδqiis a symmetry provided

δL=

∑

i

(

∂L

∂qi

δqi+

∂L

∂q ̇i

δq ̇i

)

=

d

dt

X(q,q ̇;t) (9.3)

Here, the variation must be carried outwithout using the Euler-Lagrange equations, and

the quantityX(q,q ̇;t) must be a function ofqi(t) and ̇qi(t) which islocal in time.